ANCF Finite Element/Multibody System Formulation of the Ligament/Bone Insertion Site Constraints

2010 ◽  
Vol 5 (3) ◽  
Author(s):  
Florentina M. Gantoi ◽  
Michael A. Brown ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Section ◽  
Multibody System ◽  
Insertion Site ◽  
Three Dimensional ◽  
Beam Element ◽  
Large Displacement ◽  
End Conditions ◽  
The Cross

The focus of this investigation is to study the mechanics of the knee joint using new ligament/bone insertion site constraint models that require the integration of multibody system and large displacement finite element algorithms. Two different sets of clamped end conditions at the ligament/bone insertion site are examined using nonlinear large displacement absolute nodal coordinate formulation (ANCF) finite elements. The first set of end conditions, called the partially clamped joint, eliminates only the translations and rotations at a point, allowing for the cross section stretch and shear at the ligament/bone connection. The second joint, called the fully clamped joint, eliminates all the translation, rotation, and deformation degrees of freedom of the cross section at the ligament/bone insertion site. In the case of the fully clamped joint, the gradient vectors do not change their length and orientation, allowing for the use of the constant strain assumptions. The partially clamped joint, on the other hand, allows for the change in length and relative orientation of the gradient vectors at the bone/ligament insertion site, leading to the cross section deformation induced by knee movements. Nanson’s formula is applied as a measure of the deformation of the cross section in the case of the partially clamped joint. In this study, the major bones in the knee joint consisting of the femur, tibia, and fibula are modeled as rigid bodies while the ligaments structures are modeled using the large displacement ANCF finite elements. Two different ANCF finite element models are developed in this investigation: the first model employs the fully parameterized three-dimensional beam element while the second model employs the three-dimensional cable element. The three-dimensional fully parameterized beam element allows for a straight forward implementation of a neo-Hookean constitutive model that can be used to accurately predict the large displacement as experienced in knee flexation and rotation. At the ligament bone insertion site, the ANCF fully parameterized beam element is used to define a fully or partially constrained joint while the ANCF cable element can only be used to define one joint type. The fully and partially clamped joint constraints are satisfied at the position, velocity, and acceleration levels using a dynamic formulation that is based on an optimum sparse matrix structure. The approach described in this investigation can be used to develop more realistic models of the knee and is applicable to future research studies on ligaments, muscles, and soft tissues. In particular, the partially clamped joint representation of the ligament/bone insertion site constraints can be used to develop improved structural mechanics models of the knee.

scholarly journals Hierarchical Beam Finite Elements Based Upon a Variables Separation Method

2016 ◽  
Vol 08 (02) ◽  
pp. 1650026 ◽  
Author(s):  
Gaetano Giunta ◽  
Salim Belouettar ◽  
Olivier Polit ◽  
Laurent Gallimard ◽  
Philippe Vidal ◽  
...  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Section ◽  
Three Dimensional ◽  
Finite Element Solution ◽  
Stress Fields ◽  
Element Solution ◽  
One Dimensional ◽  
Kinematic Approximation ◽  
The Cross

A family of hierarchical one-dimensional beam finite elements developed within a variables separation framework is presented. A Proper Generalized Decomposition (PGD) is used to divide the global three-dimensional problem into two coupled ones: one defined on the cross-section space (beam modeling kinematic approximation) and one belonging to the axis space (finite element solution). The displacements over the cross-section are approximated via a Unified Formulation (UF). A Lagrangian approximation is used along the beam axis. The resulting problems size is smaller than that of the classical equivalent finite element solution. The approach is, then, particularly attractive for higher-order beam models and refined axial meshes. The numerical investigations show that the proposed method yields accurate yet computationally affordable three-dimensional displacement and stress fields solutions.

scholarly journals Three-dimensional nonlinear displacement-based beam element for members with asymmetric thin-walled sections

2020 ◽  
Author(s):  
Xinlong Du ◽  
Jerome Hajjar
Keyword(s):  
Cross Section ◽  
Stiffness Matrix ◽  
Three Dimensional ◽  
Beam Element ◽  
Truss Structures ◽  
Basic System ◽  
Shear Center ◽  
Torsional Deformation ◽  
Thin Walled ◽  
The Cross

Asymmetric thin-walled sections such as steel angles and tees are widely used in truss structures and transmission towers. To address extreme limit states that these structures encounter due to extreme events such as hurricanes and earthquakes, it is important to capture their response due to large deformations caused by static or dynamic loading. In the nonlinear large deformation regime, these members have coupled axial-flexural-torsional deformation due to the so-called Wagner effect and the noncoincident shear center and centroid. A three-dimensional corotational total Lagrangian beam element is formulated and implemented in the OpenSees corotational framework to account for these coupling effects by invoking Green-Lagrange strains referenced to a basic system. In the basic system, shear forces and torque are defined with respect to the shear center, axial force is referred to the centroid, and flexure is defined around the section principle axes but in the planes containing the shear center. The element tangent stiffness matrix is derived through linearization of the governing equation obtained from the principle of virtual work. Cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation are adopted in the development. Before conducting the corotational transformation, all element end forces and displacements are transformed to act about the shear center. In order to remedy membrane locking in the inextensional bending mode, the high order bending terms in the axial strain are replaced by a constant effective strain. Cyclic material nonlinearity is considered by discretizing the cross section into a grid of fibers, tracking the steel uniaxial stress-strain constitutive at each fiber, and performing numerical integration over the cross section to obtain the section stiffness matrix. The formulation is compared against a set of experimental and numerical results to validate that the element can model geometric and material nonlinearities accurately and efficiently.

Use of General Nonlinear Material Models in Beam Problems: Application to Belts and Rubber Chains

2010 ◽  
Vol 5 (2) ◽  
Author(s):  
Luis G. Maqueda ◽  
Abdel-Nasser A. Mohamed ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Large Deformation ◽  
Cross Section ◽  
Multibody System ◽  
Three Dimensional ◽  
Nonlinear Material ◽  
Tracked Vehicle ◽  
Material Models ◽  
Belt Drives ◽  
Linear And Nonlinear

Accurate modeling of many engineering systems requires the integration of multibody system and large deformation finite element algorithms that are based on general constitutive models, account for the coupling between the large rotation and deformation, and allow capturing coupled deformation modes that cannot be captured using beam formulations implemented in existing computational algorithms and computer codes. In this investigation, new three-dimensional nonlinear dynamic rubber chains and belt drives models are developed using the finite element absolute nodal coordinate formulation (ANCF) that allows for a straight forward implementation of general linear and nonlinear material models for structural elements such as beams, plates, and shells. Furthermore, this formulation, which is based on a more general kinematic description, can be used to predict the cross section deformation and its coupling with the extension and bending of the belt drives and rubber chains. The ANCF cross section deformation results are validated by comparison with the results obtained using solid finite elements in the case of a simple tension test problem. The effect of the use of different linear and nonlinear constitutive laws in modeling belt drive mechanisms is also examined in this investigation. The finite element formulation presented in this paper is implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing detailed models of mechanical systems subject to general loading conditions, nonlinear algebraic constraint equations, and arbitrary large displacements that characterize belt drives and tracked vehicle dynamics. The successful integration of large deformation finite element and multibody system algorithms is shown to be necessary in order to be able to study the dynamics of complex tracked vehicles with rubber chains. A computer simulation of a three-dimensional multibody tracked vehicle model that consists of twenty rigid bodies and two flexible rubber chains is used in order to demonstrate the use of the formulations presented in this investigation.

Advanced Beam Theory for Multibody Dynamics

2013 ◽  
Author(s):  
Olivier A. Bauchau ◽  
Shilei Han
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Three Dimensional ◽  
Beam Theory ◽  
Bernoulli Beam ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Bernoulli Beam Theory ◽  
The Cross ◽  
Euler Bernoulli Beam

In flexible multibody systems, many components are often approximated as beams or shells. More often that not, classical beam theories, such as Euler-Bernoulli beam theory, form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to a very simple kinematic representation of the problem: the beam’s section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from Saint-Venant’s theory for torsion that the cross-section will warp under torque, leading to a three-dimensional deformation state that generates a complex stress state. To overcome this problem, sectional stiffnesses are computed based on sophisticated mechanics of material theories that evaluate the complete state of deformation. These sectional stiffnesses are then used within the framework of an Euler-Bernoulli beam theory based on far simpler kinematic assumptions. While this approach works well for simple cross-sections made of homogeneous material, very inaccurate predictions result for realistic sections, specially for thin-walled beams, or beams made of anisotropic materials. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The only approximation is that inherent to the finite element discretization. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions, as expected from Saint-Venant’s principle.

Hybridizing semianalytical and conventional finite-element schemes for simulations of electromagnetic borehole resistivity measurement

Geophysics ◽  
2017 ◽  
Vol 82 (1) ◽  
pp. E17-E26 ◽  
Author(s):  
Jiefu Chen ◽  
Shubin Zeng
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Cross Section ◽  
Resistivity Measurement ◽  
Longitudinal Direction ◽  
Integration Scheme ◽  
The Cross ◽  
Semianalytical Method ◽  
Resistivity Logging ◽  
Finite Element Schemes

A semianalytical finite-element method (FEM) has been developed to simulate electromagnetic borehole resistivity measurements in a layered underground formation. A piecewise homogeneous structure is divided into several layers. Each layer is uniform in the longitudinal direction, and the distributions of geometry and material can be arbitrary on the transverse plane, or cross section, of the layer. To develop this semianalytical finite-element scheme, the standard functional corresponding to the vector wave equation is cast to a new form in the Hamiltonian system based on dual variables that are the transverse components of electric and magnetic fields on the cross section of the layer. The 2D finite elements are used to discretize the cross section, and a high-precision integration scheme based on the Riccati equations is used to exploit the longitudinal homogeneity in the layer. By transforming a 3D layered problem into a series of 2D problems, this semianalytical FEM can save a great amount of computational costs and meanwhile achieve a higher level of accuracy when compared with conventional finite-element schemes. The flexibility of this semianalytical method can be greatly increased by hybridization with conventional finite elements, and this strategy works well for layered structures with local inhomogeneities such as borehole washouts. Several tests, including near-bit resistivity measurement and wave propagation resistivity logging, verified the effectiveness of this semianalytical FEM.

Use of General Nonlinear Material Models in Beam Problems: Application to Belt and Rubber Chains

2009 ◽  
Author(s):  
Luis G. Maqueda ◽  
Abdel-Nasser A. Mohamed ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Large Deformation ◽  
Cross Section ◽  
Multibody System ◽  
Three Dimensional ◽  
Nonlinear Material ◽  
Tracked Vehicle ◽  
Material Models ◽  
Belt Drives ◽  
Linear And Nonlinear

Accurate modeling of many engineering systems requires the integration of multibody system and large deformation finite element algorithms that are based on general constitutive models, account for the coupling between the large rotation and deformation, and allow capturing coupled deformation modes that cannot be captured using beam formulations implemented in existing computational algorithms and computer codes. In this investigation, a new nonlinear finite element dynamic model for the analysis of three-dimensional rubber chains and belt drives is developed using the finite element absolute nodal coordinate formulation (ANCF) that allows for a straight forward implementation of general linear and nonlinear material models for structural elements such as beams, plates and shells. Furthermore, this formulation, which is based on a more general kinematic description, can be used to predict the cross section deformation and its coupling with the extension and bending of the belt drives and rubber chains. The ANCF cross section deformation results are validated by comparison with the results obtained using solid finite elements in the case of a simple tension test problem. The effect of the use of different linear and nonlinear constitutive laws in modeling belt drives mechanism is also examined in this investigation. The finite element formulation presented in this paper is implemented in a general purpose three-dimensional flexible multibody algorithm that allows for developing detailed models of mechanical systems subject to general loading conditions, nonlinear algebraic constraint equations, and arbitrary large displacements that characterize belt drives and tracked vehicle dynamics. The successful integration of large deformation finite element and multibody system algorithms is shown to be necessary in order to be able to study the dynamics of complex tracked vehicles with rubber chains. A computer simulation of a three-dimensional multibody tracked vehicle model that consists of twenty rigid bodies and two flexible rubber chains is used in order to demonstrate the use of the formulations presented in this investigation.

Three-Dimensional Beam Theory for Flexible Multibody Dynamics

2014 ◽  
Vol 9 (4) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Shilei Han
Keyword(s):  
Finite Element ◽  
Cross Section ◽  
Three Dimensional ◽  
Beam Theory ◽  
Bernoulli Beam ◽  
Finite Element Discretization ◽  
Element Discretization ◽  
Bernoulli Beam Theory ◽  
The Cross ◽  
Euler Bernoulli Beam

In multibody systems, it is common practice to approximate flexible components as beams or shells. More often than not, classical beam theories, such as the Euler–Bernoulli beam theory, form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to simple kinematic representations of the problem: the beam's section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such an approach is capable of accurately capturing the kinetic energy of the system, it cannot adequately represent the strain energy. For instance, it is well known from Saint-Venant's theory for torsion that the cross-section will warp under torque, leading to a three-dimensional deformation state that generates a complex stress state. To overcome this problem, sectional stiffnesses are computed based on sophisticated mechanics of material theories that evaluate the complete state of deformation. These sectional stiffnesses are then used within the framework of a Euler–Bernoulli beam theory based on far simpler kinematic assumptions. While this approach works well for simple cross-sections made of homogeneous material, inaccurate predictions may result for realistic configurations, such as thin-walled sections, or sections comprising anisotropic materials. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The only approximation is that inherent to the finite element discretization. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions, as expected from Saint-Venant's principle.

Long waves in a straight channel with non-uniform cross-section

2015 ◽  
Vol 770 ◽  
pp. 156-188 ◽  
Author(s):  
Patricio Winckler ◽  
Philip L.-F. Liu
Keyword(s):  
Boundary Layer ◽  
Wave Propagation ◽  
Cross Section ◽  
Three Dimensional ◽  
Channel Cross Section ◽  
Cross Sectional ◽  
New Model ◽  
One Dimensional ◽  
Uniform Channel ◽  
The Cross

A cross-sectionally averaged one-dimensional long-wave model is developed. Three-dimensional equations of motion for inviscid and incompressible fluid are first integrated over a channel cross-section. To express the resulting one-dimensional equations in terms of the cross-sectional-averaged longitudinal velocity and spanwise-averaged free-surface elevation, the characteristic depth and width of the channel cross-section are assumed to be smaller than the typical wavelength, resulting in Boussinesq-type equations. Viscous effects are also considered. The new model is, therefore, adequate for describing weakly nonlinear and weakly dispersive wave propagation along a non-uniform channel with arbitrary cross-section. More specifically, the new model has the following new properties: (i) the arbitrary channel cross-section can be asymmetric with respect to the direction of wave propagation, (ii) the channel cross-section can change appreciably within a wavelength, (iii) the effects of viscosity inside the bottom boundary layer can be considered, and (iv) the three-dimensional flow features can be recovered from the perturbation solutions. Analytical and numerical examples for uniform channels, channels where the cross-sectional geometry changes slowly and channels where the depth and width variation is appreciable within the wavelength scale are discussed to illustrate the validity and capability of the present model. With the consideration of viscous boundary layer effects, the present theory agrees reasonably well with experimental results presented by Chang et al. (J. Fluid Mech., vol. 95, 1979, pp. 401–414) for converging/diverging channels and those of Liu et al. (Coast. Engng, vol. 53, 2006, pp. 181–190) for a uniform channel with a sloping beach. The numerical results for a solitary wave propagating in a channel where the width variation is appreciable within a wavelength are discussed.

Sign in / Sign up

Export Citation Format

Share Document